On graded Lie algebras of characteristic three with classical reductive null component

Abstract

Abstract We consider finite-dimensional irreducible transitive graded Lie algebras L = ∑ i = − q r L i over algebraically closed fields of characteristic three. We assume that the null component L 0 is classical and reductive. The adjoint representation of L on itself induces a representation of the commutator subalgebra L 0 ′ of the null component on the minus-one component L − 1 . We show that if the depth q and height r of L are both greater than one, then this representation must be restricted.